One definition says that $\newcommand{e}{\mathrm e}a^b = \e^{\log a^b} = \e^{b \log a}$, but this is only valid for $a \gt 0$ in the real numbers, because $\log a$ is defined only for $a \gt 0$. However, we can assign a value to $0^b$ for positive rational numbers $b$, by using (and extending) the definition that $a^b = \underbrace{a \times a \times \ldots \times a}_{b \, \text{terms}}$ where $b$ is a natural number.

And by the use of the limit definition of real numbers, we then arrive at $0^b=0$ for all positive real numbers $b$. In fact we have $\lim \limits_{a \to 0} a^b = 0$ and $\lim \limits_{x \to b} a^x = 0$ both for all positive $b$.

We might think that this idea might be of use in the complex plane too. But when we write $b=u+iv$ we get $a^b= \e^{(u+iv)\log a} = \e^{u\log a}\e^{iv \log a}$